Publication: Haar Frame Characterizations of Besov–Sobolev Spaces and Optimal Embeddings into Their Dyadic Counterparts
Authors
Gustavo Garrigós, Andreas Seeger, Tino Ullrich
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DOI
https://doi.org/10.1007/s00041-023-10013-7
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Abstract
We study the behavior of Haar coefficients in Besov and Triebel-Lizorkin spaces on R , for a parameter range in which the Haar system is not an unconditional basis. First, we obtain a range of parameters, extending up to smoothness s<1, in which the spaces F(s,p,q) and B(s,p,q) are characterized in terms of doubly oversampled Haar coefficients (Haar frames). Secondly, in the case that 1/p<s<1 and f in B(s,p,q), we actually prove that the usual Haar coefficient norm of { 2^j(f,h_{j,m}) } in b(s,p,q) remains equivalent to the norm of f in B(s,p,q), i.e., the classical Besov space is a closed subset of its dyadic counterpart. At the endpoint case s=1 and q=infty, we show that such an expression gives an equivalent norm for the Sobolev space W(1,p) in R , for 1<p<infty, which is related to a classical result by Bočkarev. Finally, in several endpoint cases we clarify the relation between dyadic and standard Besov and Triebel-Lizorkin spaces.
Citation
J Fourier Anal Appl 29, 39 (2023)
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